Question

If the angle between two lines is $\frac{\pi}{4}$ and the slope of one of the lines is $\frac{1}{2}$, then the slope of the other line is

• A. $3$ or $-\frac{1}{3}$
• B. $4$ or $-\frac{1}{4}$
• C. $2$ or $-\frac{1}{2}$
• D. $3$ or $-3$

Hint:

Notice that the two resulting values for the slope ($3$ and $-\frac{1}{3}$) are negative reciprocals of each other ($3 \times -\frac{1}{3} = -1$). This is a neat geometric property: the two possible second lines are always perpendicular to each other because they are symmetrically tilted at $45^\circ$ on either side of the first line!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the acute angle $\theta = \frac{\pi}{4}$ ($45^\circ$) between two intersecting lines, along with the slope of the first line ($m_1 = \frac{1}{2}$). We need to find the possible values for the slope of the second line ($m_2$).

Step 2: Key Formula or Approach:
The standard formula relating the acute angle $\theta$ between two intersecting lines to their slopes $m_1$ and $m_2$ is: $$\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$$ Removing the absolute value bars introduces a $\pm$ sign on the opposite side of the equation.

Step 3: Detailed Explanation:
Substitute the given values $\theta = 45^\circ$ and $m_1 = \frac{1}{2}$ into the angle formula: $$\tan(45^\circ) = \left| \frac{\frac{1}{2} - m_2}{1 + \frac{1}{2}m_2} \right|$$ Since $\tan(45^\circ) = 1$, we can simplify the expression inside the absolute value: $$\left| \frac{1 - 2m_2}{2 + m_2} \right| = 1 \implies \frac{1 - 2m_2}{2 + m_2} = \pm 1$$ This yields two separate algebraic cases to solve for $m_2$:

• Case 1: Positive Sign (+1) $$\frac{1 - 2m_2}{2 + m_2} = 1 \implies 1 - 2m_2 = 2 + m_2$$ $$-1 = 3m_2 \implies m_2 = -\frac{1}{3}$$

• Case 2: Negative Sign (-1) $$\frac{1 - 2m_2}{2 + m_2} = -1 \implies 1 - 2m_2 = -(2 + m_2)$$ $$1 - 2m_2 = -2 - m_2 \implies 3 = m_2 \implies m_2 = 3$$
Thus, the possible values for the slope of the second line are $3$ or $-\frac{1}{3}$.

Step 4: Final Answer:
The slope of the other line is $3$ or $-\frac{1}{3}$, which corresponds to option (A).